self-organization of shear bands in titanium and ti–6al–4v...

Acta Materialia 50 (2002) 575–596www.elsevier.com/locate/actamat

Self-organization of shear bands in titanium andTi–6Al–4V alloy

Q. Xue, M.A. Meyers*, V.F. NesterenkoDepartment of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA

92093-0411, USA

Received 28 May 2001; received in revised form 10 September 2001; accepted 10 September 2001

Abstract

The evolution of multiple adiabatic shear bands was investigated in commercially pure titanium and Ti–6Al–4Valloy through the radial collapse of a thick-walled cylinder under high-strain-rate deformation (�104 s�1). The shear-band initiation, propagation, as well as spatial distribution were examined under different global strains. The shearbands nucleate at the internal boundary of the specimens and construct a periodical distribution at an early stage. Theshear bands are the preferred sites for nucleation, growth, and coalescence of voids and are, as such, precursors tofailure. The evolution of shear-band pattern during the deformation process reveals a self-organization character. Thedifferences of mechanical response between the two alloys are responsible for significant differences in the evolutionof the shear band patterns. The number of shear bands initiated in Ti (spacing of 0.18 mm) is considerably larger thanin Ti–6Al–4V (spacing of 0.53 mm); on the other hand, the propagation velocity of the bands in Ti–6Al–4V (�556m/s) is approximately three times higher than in Ti (�153 m/s). The experimental shear-band spacings are comparedwith theoretical predictions that use the perturbation analysis and momentum diffusion; the shortcomings of the latterare discussed. A new model is proposed for the initiation and propagation that incorporates some of the earlier ideasand expands them to a two-dimensional configuration. The initiation is treated as a probabilistic process with a Weibulldependence on strain; superimposed on this, a shielding factor is introduced to deal with the deactivation of embryos.A discontinuous growth mode for shear localization under periodic perturbation is proposed. The propagating shearbands compete and periodically create a new spatial distribution. 2002 Published by Elsevier Science Ltd on behalfof Acta Materialia Inc.

Keywords:Shear bands; Self-organization; Titanium; Ti-6Al-4V alloy

1. Introduction

Shear localization is an important and oftendominating deformation and failure mechanism at

* Corresponding author. Tel.:+1-858-534-4719; fax:+1-858-534-5698.

E-mail address:[email protected] (M. Meyers).

1359-6454/02/$22.00 2002 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc.PII: S1359 -6454(01 )00356-1

high strain rates e.g. [1] and [2]. It has been theobject of numerous studies, focused both on themechanics and microstructural aspects of the pro-cess. The constitutive equations governing adia-batic shear localization are well known, and theapproaches developed by Clifton [3], Bai [4], Mol-inari and Clifton [5], Wright [6], and Wright andWalter [7] successfully predict the strains for the

576 Q. Xue et al. / Acta Materialia 50 (2002) 575–596

initiation as well as other parameters, such as bandthickness. These analyses have been corroboratedby controlled, instrumented experiments; promi-nent among those are the pioneering results byMarchand and Duffy [8]. From a microstructuralevolution viewpoint, a number of structural alter-ations, including dynamic recovery, dynamicrecrystallization, phase transformations, andmelting/resolidification have been observed. Fortitanium and titanium alloys, Schechtman et al. [9],Grebe et al. [10], Meyers and Pak [11], Meyers etal. [12], and Timothy and Hutchins [13,14], haveidentified important microstructural alterations.Ramesh and coworkers [15–17] have recentlyinvestigated Ti and Ti alloys.

However, the great majority of studies addressedan individual shear band. Exceptions are the stud-ies by Bowden [18] and Shockey [19] showingmultiple shear bands and emphasizing their spac-ing. It was shown by Nesterenko et al. [20] thatshear bands in titanium exhibit a self-organizationbehavior with a characteristic spacing. Theseobserved spacings were compared with the threeexisting Grady–Kipp [21] and Wright–Ockendon[22] and Molinari [23] theories. One of the objec-tives of this report is to expand on the initial find-ings [20] and to propose a new, two-dimensionaltreatment for the self-organization of shear bands.The role played by shear bands in the initiation andevolution of damage is also characterized.Titanium and Ti–6Al–4V alloy were chosenbecause they both have close compositions, den-sity, and a demonstrated propensity for shear local-ization but have rather different mechanicalproperties; the yield strength of Ti–6Al–4V isapproximately twice that of Ti.

2. Theoretical predictions

A few theoretical predictions for spacing ofshear bands have been proposed to characterize thespatial distribution of shear bands. Since the evol-ution of multiple shear bands is very complex dueto the interactions among them, most analyses arestill simplified under one-dimensional simpleshear condition.

Grady [24] was the first to propose a pertur-

bation solution to shear instability of brittlematerials. The governing equations were simplifiedas a dimensionless system. The principle of pertur-bation analysis is shown in Fig. 1. The schematicprocess reveals the development of perturbationsin an initially homogeneous deformation. The com-petition among small perturbations produces a newdistribution of perturbations with larger ampli-tudes. In Fig. 1, two wavelengths (L1 and L3) areshown. The amplitude of the perturbation withlarger wavelength (L3) grows faster and dominatesthe process at t3. Shear bands evolve from thegrowth of perturbations. A Newtonian viscousconstitutive equation was used to describe theshear deformation of the solid medium:

t � h(T)∂v∂y

, (1)

where t is the shear stress, v is the velocity, and theviscous coefficient h(T) is temperature dependent:

h(T) � h0 exp[�a(T�T0)], (2)

h0 and T0 are a material constant and the referencetemperature, respectively. The perturbation wave-length associated with the instability correspondsto the minimum spacing. The following character-istic wavelength was obtained from the pertur-bation analysis:

LG �2pg0� kCa2h0

�1/4

. (3)

Fig. 1. Schematic of evolution perturbations with develop-ment of characteristic spacing for subsequent shearbandnucleation.

577Q. Xue et al. / Acta Materialia 50 (2002) 575–596

Grady and Kipp [21] later proposed anotherapproach (called here the GK model) to determinespatial distribution of shear bands. They extendedMott’s [25] early analysis for dynamic fracture.Mott [25] pointed out that the velocity of stressrelease away from the point of fracture was con-trolled by momentum diffusion and was muchlower than elastic wave velocity. Fig. 2 gives thedeformation distribution under one-dimensionalsimple shear and shows in a schematic manner theprinciple of momentum diffusion. The momentumdiffusion due to the unloading creates a rigidregion between the shear band and the plasticdeformation region. The rigid–plastic interfacepropagates at a speed lower than the elastic wavevelocity. A linear relaxation of stress was used todescribe approximately the relation of unloadingstress. A simple constitutive equation, t = t0[1�a(T�T0)], was applied; a is a softening parameter.Work hardening and strain rate sensitivity are neg-lected. Grady and Kipp [21] indicated that theshear band spacing should correspond to the mini-mum localization time during which the shear band

Fig. 2. Structure of one-dimensional shear band and stressrelease behavior in the vicinity of the shear band. Note that rigidregions and plastic flow regions are separated by a propagatinginterface (adapted from Grady and Kipp [21]).

reached its critical width. The predicted spacing,LGK, is:

LGK � 2� 9kCg30a2t0

�1/4

. (4)

Wright and Ockendon [22] developed theirtheoretical model (WO model), based on the analy-sis of small perturbations. The basic concept issimilar to Grady’s earlier work [24]. The followingconstitutive equation for a rate dependent materialwas used:

t � t0[1�a(T�T0)]� gg0�m

, (5)

where m is the strain rate sensitivity. The spacingof shear bands, LWO, is:

LWO � 2p�m3kCg30a2t0

�1/4

. (6)

More recently, Molinari [23] modified the WOmodel by adding strain hardening:

t � m0(g � gi)ng mTn. (7)

m0 and n are constants and gi is a pre-strain, andm is the strain-rate sensitivity and n is the strainhardening exponent. The predicted spacing ofshear bands is:

LM �2px0�1 �

34

rc∂g∂g

bt0∂g∂T

��1

, (8)

where x0 is the wave number. For a non-hardeningmaterial, the Molinari model takes the simpleform:

LM� � 2p�kCm3(1�aT0)2

(1 � m)g30a2t0�1/4

. (9)

Eq. (7) for linear thermal softening can be rewrit-ten as:

g � t1/m[m0(1�aT)]�1/m(g � gi)�n/m. (10)

The initial wave number x0 is obtained directlyfrom the non-hardening solution of shear bandspacing:


x0 �2pLM�

. (11)

The spacing of shear band with work hardeningeffect is obtained by substituting Eqs. (8) and (10)into Eq. (11):

LM � �1�34rcbt20

n(1�aT)bag ��1

· �kCm3(1�aT0)2

(1 � m)g30a2t0�1/4

, (12)

where the pre-strain is assumed to be zero, gi=0.The above predictions are classified into two

types: momentum diffusion to describe the laterstage of the shear localization [21], and the pertur-bation analysis to explain the initial growth ofshear bands at an early stage [22–24]. They canalso be expressed as:

Grady � Kipp

L � 2p� kCg3a2t0

�1 /4

·91/4

p(13)

Wright � Ockendon

L � 2p� kCg3a2t0

�1 /4

·m3/4 (14)

Molinari

L � 2p� kCg3t0a2�1 /4

· �m3(1�aT0)2

(1 � m) �1 /4

for n � 0. (15)

The WO and Molinari models are the same exceptfor a factor [(1�aT0)2 /(1 � m)]1/4. Since m is muchless than 1 and T0 is the reference temperature, thisfactor is approximated as (1�aT0)1/2. The form ofthe equations enables direct comparison between thethree predictive models. Generally speaking, if thestrain hardening effect is ignored, the Molinari pre-diction is of the same order as the WO model, exceptfor the coefficient. The coefficient in the GK modelis independent of work hardening. Under such anextreme condition, the absence of strain-rate sensi-tivity in the GK model does not affect the distri-bution of shear bands, although we can incorporatethe strain rate term into the constitutive equation byobtaining t0 from Eq. (7). The coefficient of GKmodel is 0.55, which is at least 5 to 10 times largerthan m3/4 since m is about 10�2 for most metals.

3. Materials and experimental procedure

3.1. Materials and constitutive behavior

Commercially pure titanium is a typical highpurity α titanium with HCP structure. Titaniumundergoes an allotropic transformation from HCP(α-Ti) to BCC (β-Ti) at about 882°C. Nesterenkoet al. [20] used this material to examine the spatialdistribution of shear bands. In the present research,the former cylindrical specimens after collapsewere re-examined for the detailed characterizationof shear localization.

Ti–6Al–4V alloy, in the form of a rod 1 inch indiameter, was used (ASTM B348-95 with grade 5).It was in the annealed condition (705°C for twohours and air cooling), providing a two-phasestructure with elongated grains. The grain size isabout 7 µm length and 3 µm width.

The constitutive response of the materials usedwas described by means of Zerilli–Armstrong (Z–A) equations. Quasi-static and dynamic tests wereconducted to obtain the Z-A parameters. The ther-mal softening parameters a0, a1, b0 and b1 weretaken directly from Zerilli and Armstrong [26].Zerilli and Armstrong extended their theoreticalprediction, initially developed for BCC [27] andFCC [28] metals, to HCP [26]. They pointed outthat HCP metals have a mechanical response fall-ing somewhere between BCC and FCC metals.They used a combination of predominant interac-tions, the Peierls stress type (BCC) and the forestdislocations type (FCC) to describe the constitutiveresponse of HCP metals:

s � sa � Be�(b0�b1 ln e)T (16)� B0eCne�(a0�a1 ln e)T.

In Eq. (16), B, B0, b0, b1, a0, a1 and Cn are para-meters. The constant sa represents the athermalcomponent of the flow stress (that includes thegrain size effect). The second and the third termsdescribe the thermal activation from the Peierlsstress interaction and the intersection of dislo-cations, respectively. The parameters not obtainedexperimentally were taken from Zerilli and Arm-strong [26], Chichili et al. [16,17], and Meyers etal. [12] for Grade 2 CP titanium. The predictionequation was fitted and the parameters are listed


Table 1Parameters of Zerilli–Armstrong prediction for Ti and Ti–6Al–4V alloy

Mixed model: s = C0 + B0eCn exp(�a0T + a1T ln e) + B exp(�b0T + b1T ln e)

C0 (MPa) B0 (MPa) a0 (K�1) a1 (K�1) Cn B (MPa) b0 (K�1) b1 (K�1)

CP Ti 0 990 1.1×10�4 7.5×10�5 0.5 700 2.24×10�3 9.73×10�5

Ti–6Al–4V 340 1135 1.4×10�5 7×10�5 0.5 1485 2.75×10�3 7.47×10�5

in Table 1. Fig. 3(a) shows a comparison of theexperimental curves by Chichili et al. [16,17] andMeyers et al. [12] with the Z-A predictions for Ti.The predicted curves are in good agreement withthe experimental data.

Fig. 3. Comparison between the experimental data and theZerilli–Armstrong predictions for (a) CP titanium. Experimentaldata from Chichili et al. [16] (quasi-static, e = 10�3 s�1;dynamic, e = 5×103 s�1, and Meyers et al. [12] (dynamic,e = 1.2×103 s�1).(b) Ti–6Al–4V. Note that ‘Q’ and ‘D’ rep-resent quasi-static and dynamic; and the corresponding strainrates are 10�3 s�1 and 3.8×103 s�1, respectively.

Ti–6Al–4V alloy includes both HCP (α phase)and BCC (β phase) structure. Its constitutiveresponse depends much more on the mechanismthat dominates BCC metals than titanium. Theparameters are listed in Table 1. The predictedresponses are compared with experimental resultsin Fig. 3(b). There is a good match with experi-mental curves. These constitutive equations wereused in the prediction of the adiabatic temperaturerise (Section 5.4).

3.2. Thick-walled cylinder explosion technique

The thick-walled cylinder implosion techniquewas introduced by Nesterenko et al. [29,30] andrepresents an improvement from the containedexploding cylinder technique developed byShockey [19]. The technique is described in detailelsewhere [31,32] and will be only brieflypresented here.

The specimen is sandwiched between a copperdriver tube and a copper stopper tube and is col-lapsed inwards during the test. OFHC copper wasused to make these tubes. The internal diametersof the inner copper tube were selected to produceprescribed and controlled final strains. In somespecial cases a central steel rod was also used. Theexplosive is axi-symmetrically placed around thespecimen. The detonation is initiated on the top.The expansion of the detonation products exerts auniform pressure on the cylindrical specimen anddrives the specimen to collapse inward. The deton-ation velocity of the selected explosive is approxi-mately 4000 m/s and the density of the explosiveis 1 g/cm3. The velocity of the inner wall of thetube was determined by an electromagnetic gage.The initial velocity of collapse of the inner tubewas found by Nesterenko et al. [33] to be approxi-mately 200 m/s.


It is possible to obtain an estimate of the upperbound of this velocity by a simple analytical calcu-lation which equates the chemical energy of theexplosive to the kinetic energy of the collapsingcylinder and of the detonation gases. This is theGurney approach. An equation applicable to thegeometry used here was developed by Meyers andWang [34]. Neglecting the effect of plastic defor-mation work, we have:

V � (17)

�2E�3

5Mc

� 2�Mc �2R � r

r�

2rR � r

�1/2

.

V is the collapse velocity of the cylinder, E is theGurney energy, M and c are the masses of metaland explosive charge, respectively, R is the exter-nal radius of the explosive charge (equal to 30mm), and r is the external radius of the cylinder(equal to 15 mm). The Gurney energy wasobtained through the empirical relationship byKennedy [35]:

√2E � D/3. (18)

D is the detonation velocity, equal to 4000 m/s.The initial collapse velocity was equal to 366 m/s.Recent calculations using RAVEN have yielded acollapse velocity of 250 m/s. These calculationsrepresent an upper bound and shows that the meas-ured value of 200 m/s is reasonable. Incorporationof the plastic deformation work would reduce thecalculated velocity obtained from Eq. (17).

To avoid extra wave reflection from the freeinterface between the stopper and the specimen, ashrink-fit technique was applied. The outer diam-eter of stopper was designed 0.03 mm larger thanthe inner diameter of the specimen. For Ti–6Al–4V alloy, special steel rods were applied to stopthe radial deformation at a much earlier stage. Steelrods with diameters of 7 mm and 9 mm were usedas the core at the center of cylinder specimen tostop the deformation.

The collapse of thick-walled cylinder specimenoccurs in a plane strain condition. The stress statecan be considered as a superposition of a hydro-static pressure and a pure shear stress due to theaxi-symmetrical geometry and loading. The

maximum shear strain occurs on the internal sur-face of the cylindrical specimen, and thus shearbands preferentially initiate there. After eachexperiment, the cylinders were sectioned, groundand polished. The lengths of shear bands, li, theedge displacements, di, the average radius of finalinternal boundary, Rf, and the angle between spatialposition from origin, �i, were measured as shownin Fig. 4. In order to compare the deformation atthe different positions on the specimen, an effec-tive strain is used as:

eef �2√3err �

2√3

ln �r0

rf�, (19)

where r0 and rf are the initial and final radii of areference point. The effective global strain at theinternal boundary of the specimen is considered asa characteristic value of deformation since all thespecimens have the same initial dimensions. Basedon the number of distinguishable shear bands, theaverage spacing between them is:

L ��i Rf

ni√2, (20)

Fig. 4. Characteristics of shear band pattern and basicmeasurement; �i is the reference angle and Rf is the final radiusof the collapsed cylinder; Li is the spacing between the ith andthe i�1 shear band, li is the length of the ith shear band anddi is its edge length.


where ni is the number of shear bands at the parti-cular region i and �i is the corresponding anglereferring to the original angle. If ni = ntotal, �i =2p. The spacing between shear bands decreases asplastic deformation proceeds due to the continuingdeformation [36]. The measured spacings are actu-ally the sum of the real developing part and thegeometrical part due to the change of the cylinderconfiguration. This geometrical effect needs to besubtracted from the results by an appropriate cor-rection. The values of spacings at the initiation ofthe bands was taken as a reference. The spacing ofshear bands can be expressed as:

L ��i Rf

ni√2 �Rf0

Rf��

i

Li,i�1

ni�Rf0

Rf�, (21)

where Rf0 is the radius of the specimen at shearband initiation, and Rf is the final radius, at anylarger effective strain. Li,i�1 is defined as the spac-ing between ith and i�1th shear bands in Fig. 4.Since Rf � Rf0, the corrected spacing based on thefinal configuration always has a lower value.

4. Microstructural characterization

There are significant differences between thetwo materials, in spite of their close compositionand thermal properties. Fig. 5 shows the character-istic configuration of shear bands for Ti and Ti–6Al–4V. Three bands are seen in Fig. 5(a), whereasonly one band is seen in Fig. 5(b); it should alsobe noted that the magnification in Fig. 5(a) isalmost twice of that in Fig. 5(b). These results sug-gest (and this will be analytically developed in Sec-tion 6) that the interaction between nuclei of shearbands in Ti–6Al–4V is relatively weak. Each shearband grows fast once it nucleates. Unloading froma developed shear band during growth reduces newnucleation sites in the surrounding areas. Theexamination of the ratio of the lengths of shearbands and their edge displacement, 1/d, also indi-cates the major difference between Ti and Ti–6Al–4V. 1/d is about 14.5 for Ti and 26.5 for Ti–6Al–4V at the same value of d. This result clearly indi-cates that the shear bands develop much faster in

Fig. 5. Comparison of nucleation sites between Ti and Ti–6Al–4V alloy. (a) Titanium and (b) Ti–6Al–4V alloy.

Ti–6Al–4V alloy than in Ti. The detailed analysisof shear band development is presented in Sec-tion 6.

The thickness of the band was found to be afunction of the distance from its tip. This holds forboth materials and had been observed earlier byNesterenko et al. [20] for Ti. The largest shear-band thickness in Ti is approximately 10 µm. Thetip of the band reaches a level of intermittent local-ization in which the grain scale plays a role. Thisis seen in Fig. 6(a) in which the band (in Ti–6Al–4V) is shown by arrows. The thickness is on theorder of 1 µm. There is a slight discontinuity,


Fig. 6. Evolution of shear band width and morphology in Ti–6Al–4V alloy: (a) fine tip section of the band; (b) well-developed section of the band.

marked by a black arrow. Fig. 5(b), on the otherhand, shows a well-developed band. The thicknessis approximately 8 µm. The interior is virtually fea-tureless, and the original microstructure has beeneliminated. It has been shown by Grebe et al. [37]for Ti–6Al–4V and by Meyers and Pak [11] andMeyers et al. [12] for Ti that the structure withinthe shear band consists, at a sufficiently high shearstrain, of small (0.1–0.3 µm) equiaxed grains. Thefeatureless material within the band was concludedto be the result of a rotational dynamic recrystalliz-ation instead of the product of phase transform-ation, as suggested by the early literature on shear

bands, based exclusively on optical microscopyobservations. It is now recognized that a numberof microstructural changes can and do occur withinshear bands, including dynamic recovery andrecrystallization, phase transformation and evenamorphization. The presence of fine equiaxedsubgrains within the shear band with the averagediameter of 0.2 µm provided a direct evidence fordynamic recrystallization. However the defor-mation time is lower, by orders of magnitude, thanthe time required for boundary migration. Thus, arotational mechanism, previously mentioned byDerby [38] in his classification of dynamic recrys-tallization, was proposed for the shear bands[39,40].

Another important feature of shear bandsobserved in the exploding cylinder geometry is thebifurcation. This bifurcation of bands is geometri-cally necessary due to the spiral trajectory of thebands, starting in the internal surface. In order toretain the same band spacing as they travel out-wards, new bands have to be generated. Bifur-cation is one possible means and it has been pre-viously observed in steels (e.g. [41], Fig. 5). Fig.7 shows a bifurcation event for Ti–6Al–4V.

Shear localization precedes failure mechanismssuch as voids and cracks. A well-developed shearband is often accompanied with voids and cracks

Fig. 7. Shear band bifurcation and induced damage in Ti–6Al–4V alloy.


within it. These defects evolve and interact withshear bands during shear localization. Defectsaccelerate localization and localized deformationcreates fresh nucleation of micro-voids and cracks.

The evolution of defects within shear bands inTi–6Al–4V alloys showed, in this geometry, anentire range of events associated with damage andfailure. The nucleation of micro-voids is con-sidered as a result of the tensile stress inside shearbands. Void evolution is comprised of the threemain stages: nucleation, growth, and coalescence.Fig. 8(a) shows nucleation stage of voids withinshear bands. Three voids with circular or ellipticalshape are initiated within shear band. All voidshave smooth surfaces. It means that the materialinside the band is quite soft and the temperaturewithin the band is high. These voids nucleate com-pletely inside the band and are surrounded by the

Fig. 8. Void nucleation and growth inside a shear band in Ti–6Al-4V alloy: (a) nucleation of voids within a shear band; (b) growthof voids; (c) elongation and rotation of voids; (d) coalescence.

flow of localized deformation. They grow up withthe localization progress in the growth stage. Mostof voids first grow to the width of the band andthen are elongated to elliptical shape along thedirection of the shear band (see Fig. 8(b)). Furthergrowth of these voids is accompanied with rotationalong the shear direction. The second stage of voidgrowth is characterized with the void elongationand rotation. Several voids are elongated to ellipseshape and rotated along direction of shear momentin Fig. 8(c). The third stage of void developmentis coalescence of voids and crack generation. Fig.8(d) shows the whole process of void coalescence.These elliptical voids tilt their long axis to a certainangle coalescence of a group of voids insideshear band.

Timothy and Hutchings [13,14] proposed a sim-ple schematic plot to express the void evolution


within shear bands. A similar schematic descrip-tion is presented in Fig. 9. Although there are somedifferences in the details, the major processes intheir plot are in good agreement with what isobserved in Ti–6Al–4V alloy represented in Fig.8. Voids nucleate within the shear bands whenthere is tension. They grow until their edges reachthe boundary of the band, where the strength of thematerial is higher due to the lower temperature.Growth continues within the bands, and the voidsbecome elongated in the process. They can eventu-ally coalesce, creating complete separation. Thisprocess in the absence of shear strains is shown inFig. 9(a). If there are shear strains present concur-rently with tension, the voids are elongated androtated. This is shown in Fig. 9(b). The ellipsoidsno longer share a common major axis. Their axesare parallel and inclined to the shear-band plane.In the present case a mixture of the two cases wasobserved. The collapsed cylinder specimens movecontinuously inward during the implosion process,and only compressive and shear stresses are gener-ated at this stage. During the unloading stages, ten-sile circumferential stresses are generated in thecylinders because of the radial dependence of thestresses. These tensile stresses are responsible forfurther growth of voids generated within the bands.

Fig. 9. Schematic representation of void nucleation, growth,and coalescence in (a) absence and (b) presence of continuingshear.

5. Shear-band spacings

5.1. Titanium

In the present study, the titanium samples fromthe experiments carried out by Nesterenko et al.[20] were carefully re-examined by using bothoptical microscopy and SEM. The configuration ofshear bands on the cross sections of the cylindricalspecimens was described before [20,34]. It consistsof spiral trajectories. The lengths, the edges andspacing of shear bands were re-measured. Detailedobservation shows that there is a large number ofsmall shear bands that had not been counted in theformer work. The total number of shear bands inCP titanium at this stage is 116, while the earliercount [20] was 11. The large difference is due tofact that emerging shear bands were neglectedearlier. At an effective strain of 0.92, the numberof shear bands is 108. The earlier count was 30.The spectra of shear-band distribution at the twostrains are shown in Fig. 10. The numbers of shearbands at both the early stage and the later stageare similar. This implies that there is no newnucleation of shear bands at the internal boundaryafter an effective strain of 0.55. Only previouslynucleated shear bands grow between eef = 0.55 andeef = 0.92. The spacing of shear bands in CPtitanium reaches a saturation state and remainsconstant. The average shear band spacing is 0.18mm. Fig. 10 shows that the number of bands grow-ing at the strain of 0.92 is smaller than at 0.55; thespacing is dependent on the size of the shear bands.This size-dependent spacing necessitates a two-dimensional treatment; this is done in Section 6.

5.2. Ti–6Al–4V alloy

Four different effective strains were selected toexamine the evolution of the patterns of shearlocalization. They are 0.13, 0.26, 0.55 and 0.92.Shear localization is initiated at a much earlierstage in Ti–6Al–4V alloy than in CP Ti. In com-parison, at the same effective strains of 0.55 and0.92 for Ti, the shear bands are much shorter [47].A well-developed shear-band pattern is alreadyformed at an effective strain of 0.26. The numbersof shear bands are: 68 for eef = 0.26, 64 for


Fig. 10. Spectra of shear band distribution in CP titanium attwo different stages: (a) eef = 0.55; and (b) eef = 0.92.

eef = 0.55, and 68 for eef = 0.92. The numbers ofshear bands remain approximately constant, indi-cating that once the shear band becomes well-developed, no further nucleation takes place. Theshear band pattern at an early stage, at an effectivestrain of eef = 0.13, gives a number of only 25. Thisstrain is within the nucleation stage.

Fig. 11 shows the distribution of shear bands fortwo effective strains in Ti–6Al–4V alloy. Thenucleation of shear bands occurs within a certainrange of strains after a critical strain is reached.Two principal differences exist between the evol-

Fig. 11. Spectra of shear bands distribution in Ti–6Al–4Valloy: (a) early stage (eef = 0.13); (b) late stage, (eef = 0.26).

ution in Ti and Ti–6Al–4V. One is that the rangeof strains for nucleation in Ti–6Al–4V alloyappears to be broader than in titanium. The otheris associated with the velocity of shear bands. Thevelocity can be estimated from the collapse velo-city of the cylinder and the length of the bands.The total travel distance of the inner surface of thecylinder divided by the collapse velocity providesthe propagation time. The calculation is carried outin Section 5.3. The spacing is, as for Ti, observedto increase with the size of the bands.


5.3. Propagation velocity of shear bands

The velocity of shear band propagation can beestimated from the experimental data of the col-lapsed cylinder tests. According to Nesterenko etal. [20,30,33], the average shear strain rate in theinternal boundary of the thick-walled cylinderspecimen is 3.5×104 s�1. The corresponding radialstrain rate, err is 1.75×104 s�1. This corresponds toan average collapse velocity of 200 m/s. Radialstrains at the two deformation stages can be trans-formed into the corresponding effective strains byusing Eq. (19). For CP Ti, the radial strains are0.476 and 0.797, corresponding to eeff = 0.55 andeeff = 0.92, respectively. For Ti–6Al–4V alloy, theyare 0.113 and 0.225, corresponding to eeff = 0.13and eeff = 0.26, respectively. The differences ofduration between two strains are:

�t ��err

err

�err2�err1

err

, (22)

where the subscripts 1 and 2 represent the twostages of strain. The differences of duration are18.3 µs for Ti and 6.46 µs for Ti–6Al–4V. Theincrements of the longest length of shear bands forthese two materials are obtained directly from thedata in Figs. 10 and 11. The maximum length ofshear bands in Ti increases from 1.2 mm(eeff = 0.55) to 4.0 mm (eeff = 0.92), while that inTi–6Al–4V increases from 0.58 mm (eeff = 0.13) to4.18 mm (eeff = 0.26). The average velocities ofshear band propagation are directly calculated andare equal to 153 m/s for Ti and 556 m/s for Ti–6Al–4V.

Zhou et al. [42,43] measured and numericallycalculated the velocity of propagation of shearband in Ti–6Al–4V impacted at 64.5 m/s. Thepropagation velocities were 50–75 m/s, one orderof magnitude less than the current results. It is con-cluded that the velocity of propagation of shearbands is dependent on the energy available forrelease at the front. The energy release rate for Ti–6Al–4V is significantly higher than for Ti due toits higher yield stress. This explains, qualitatively,its higher propagation velocity.

Mercier and Molinari [44] proposed a theoreticalanalysis for the velocity of propagation of shearbands incorporating some of these concepts. The

band is evaluated in simple shear (Mode II). Fig.12(a) shows the Mode II configuration used byMercier and Molinari [44]. A slab with thickness2h is sheared with top and bottom boundaries dis-placed at velocities C and �C, respectively. Ashear band propagates in the center of the slab withvelocity V. The band thickness is t and the processzone size is l. Using a variational method, twovariational relations were obtained as

h

�h

[sxx]��dy �

�

��

[sxy]h�h (23)

� � �

��

h

�h

�sij

∂eij∂V

� rai

∂vi

∂V�dx dy � 0,

Fig. 12. Calculated propagation velocity for shear band, usingMercier and Molinari’s [42] theory): (a) velocity configurationused in calculation; (b) predicted velocities for different con-ditions.


� �

��

h

�h

�sij

∂eij∂l

� rai

∂vi

∂l�dx dy � 0, (24)

where ai = vi,kvk (i = 1,2) is the acceleration ofshear bands, and sxy are the deviatoric Cauchystresses. These relations were further simplifiedunder the constrained condition. The velocity ofshear band V and the characteristic length of theprocess zone l can be obtained by the solution offollowing equations:

�

��

[sxy(x,h) � sxy(x,�h)�2sxy(x,0)]dx (25)

�rCV(h�t) � 0,

�2 � �

��

h

�h

�x�sxx

∂exx

∂x� sxy

∂exy

∂x �� sxxexx � sxy

∂vy

∂x �dx dy (26)

� � �

��

h

�h

rxV��∂vx

∂x �2

� �∂vx

∂y �2�dx dy

� 0.

For a rigid perfectly plastic material, anexpression of the form was obtained by Mercierand Molinari [44]:

V �sf

rCg�lh,

th�, (27)

where sf is the flow stress, r is the density. Thus,the shear band velocity is directly proportional tothe flow stress. The value for the function g(l/h,t/h) = 0.343 was obtained from Fig. 5 of Mercierand Molinari [44]. The predicted velocities areshown in Fig. 12(b). The velocity C was taken, toa first approximation, as 200 m/s. This is the initialcollapse velocity for the cylinder. For the currentresults, the predicted velocities are 190.5 for Ti (sf

= 500 MPa) and 571.5 m/s for Ti–6Al–4V (sf =1500 MPa), respectively. The correspondingexperimental velocities are 153 and 556 m/s,respectively. The calculated and experimentally

inferred results are in excellent agreement. Thevariational theory of Mercier and Molinari [44]predicts correctly the dependence of propagationvelocity on flow stress.

5.4. Comparison of predicted and experimentalresults

The significant differences in shear-band spacingencountered in the two materials is surprising. Itis indeed interesting to notice that these twomaterials, with fairly close compositions based onTi, have such a dissimilar response. The physicaland mechanical parameters for CP Ti and Ti–6Al–4V alloy are listed in Table 2. Using these para-meters, the predicted spacings of shear bands fromthe three theoretical models presented in Section 2can be calculated. The predictions for the threemodels are shown in Table 3. Since the Grady–Kipp model does not consider the strain rate effect,the shear yield stresses in this model correspondto those under dynamic loading condition. The WOand Molinari (without work hardening) modelsgive predictions for Ti as Li = 0.29 mm and Li

= 0.24 mm, respectively. The experimental resultsshow that the spacing of shear bands is 0.18 mm.The predictions are reasonably close to theobserved spacing. The GK model predicts the shearband spacing to be 2.13 mm, which is much larger.Thus, the predictions of WO/Molinari models arein approximate agreement with the observed shearband spacing in CP titanium. If work hardening isincorporated into the Molinari model, the predictedspacing increases somewhat. The Molinari equ-ation is applied in its simplified, non-work harden-ing form (Eq. (15)) and in the form given below,which uses a softening term different from theoriginal formulation for ease of mathematicalmanipulation. The following constitutive equationwas used:

t � m0(g � gi)n g m(1�aT). (28)

The corresponding spacing is

LM � L0�1�34rcbt0

n(1�aT)ag ��1

, (29)

L0 is the spacing for the non-work hardening case.


Table 2Physical and mechanical parameters for prediction of shear-band spacinga

Material Density Heat capacity Thermal Thermal softening Strain rateconductivity factor hardening index

r (g/cm3) C (J/kg K) k (W/mK) a (K�1) M

Titanium 4.51 528 16.44 10�3 0.027Ti–6Al–4V 4.43 564 3.07 10�3 0.017

Strain hardening Shear stress Dynamic shear Shear strain rateindex stressn (MPa) (MPa) g (s�1)

Titanium 0.18 180 280 6×104

Ti–6Al–4V 0.15 490 650 6×104

a Thermal properties and densities are from Ref. 50; thermal softening factor from Nesterenko et al. [20]; other parameters fromexperimental data.

Table 3Comparison of shear band spacings between the experimental results and theoretical predictions

Spacing (mm) Experimental data Grady–Kipp Wright–Ockendon Molinari (without Molinari (withstrain hardening) strain hardening

effect)

CP titanium 0.18 2.13 0.29 0.24 0.64Ti–6Al–4V alloy 0.53 1.15 0.10 0.09 0.10

The predicted spacing is increased from 0.24 mmto 0.64 mm.

The number of shear bands for Ti–6Al–4V alloyis essentially constant for the global effectivestrains 0.26, 0.55 and 0.92; an average number canbe taken as approximately 66. The correspondingspacing of shear bands is Li = 0.526 mm. There-fore, it can be concluded that after this spacing isreached, no further nucleation occurs. The pre-dicted results for shear band spacing are alsoshown in Table 3. The values of spacing for WOand Molinari models are 0.10 mm and 0.09 mm,respectively. Thus, the experimental results areroughly 5 times the theoretical predictions. On theother hand, the GK model predicts a spacing L =1.15 mm, which is roughly double the experi-mental results. So, the experimental and calculatedspacings do not show as good an agreement as inTi. Moreover, the proposed models do not predictobserved tendency in shear band spacing betweentitanium and Ti–6Al–4V.

The failure of these theoretical predictions in Ti–6Al–4V alloy suggests that some mechanisms ofshear band development have not been includedinto the current models. Considering the Molinarimodel with work hardening, the strain hardeningis 0.15 for the Ti–6Al–4V alloy, which is lowerthan the one for Ti. The incorporation of this strainhardening index cannot result in a substantialchange of the spacing of shear bands. Therefore,strain hardening should not be the main reason forthe failure of the prediction.

The Grady–Kipp model overestimates the spac-ing considerably, whereas WO and M modelsunderestimate it. Section 6 will present a two-dimensional model in which shielding is incorpor-ated into the nucleation.

The temperature increment due to deformationcan be estimated through the predictive constitut-ive relations. This temperature rise helps to explainsome of the differences in shear bands between thetwo materials. The temperature change is given by:


dTdt

� bserC(T)

, (30)

T

T0

rC(T)bs(T)

dT � t

0

edt � e

0

de � e. (31)

The stress as the function of temperature, s(T), isobtained directly from the Zerilli–Armstrong equ-ation (Eq. (16)). The heat capacities C(T) are:

CP Ti C(T) � 0.514 � 1.357×10�4 T

�3.366×103

T2

� 2.767×10�8 T2[J /kgK] (32)

Ti�6Al�4V C(T) � 0.559

� 1.357×10�4 T�3.366×103

T2

� 2.767×10�8 T2[J /kgK] (33)

In the adiabatic case (b = 1), numerical integrationof Eq. (31) gives the relation between strain andtemperature increment. Fig. 13(a) shows the tem-perature as a function of effective strain for the twomaterials. The temperature rises faster for Ti–6Al–4V. The effective strains corresponding to theinitiation of localization, approximately 0.5 and0.3, correspond to the same temperature rise of 100K. A greater plastic deformation is needed toincrease the temperature to the same level in Ti.Culver [45] introduced a simple relation betweentrue strain and engineering shear strain.

e � ln�1 � g �g2

2(34)

g � √2e2e�1�1 (35)

It should be mentioned that his conversion relationis for simple shear case, although, strictly speak-ing, the state of stress in these experiments is ofpure shear. The relations between temperature andthe converted engineering shear strain are plottedin Fig. 13(b). The temperature corresponding toTm/2 (one half the melting temperature) is anapproximate measure of the onset of recrystalliz-ation. It is marked in Fig. 13(b). It represents shearstrains of 5 and 3 for Ti and Ti–6Al–4V, respect-ively. The shear strains in the bands can be esti-

Fig. 13. Predicted temperature for CP titanium and Ti–6Al–4V alloy (a) as a function of effective strain g, and (b) as afunction of shear strain.

mated from the edge length/thickness (d/t) ratios.The typical edge length in the Ti–6Al–4V speci-men at a global strain of 0.13 is about 30 µm ateef = 0.26 and the typical thickness of the shearband is 8 µm. This gives a shear strain about 4.At a more advanced stage (global strains of 0.26and 0.55) the edge lengths are about 0.33 mm and1.67 mm, respectively. These correspond to strainsof 41 and over 200, respectively. These are vastlysuperior to the recrystallization temperature andcould even generate melting.


6. Two-dimensional effects among multipleshear bands

The failure of the one-dimensional models toexplain the evolution of shear-band spacing led tothe considerations outlined below. In order todescribe such self-organization behavior two basicprocesses, initiation and growth, need to beincluded separately. Initiation can be treated as theselective activation of sites. Growth represents thecompetition among bands and interaction duringthe propagation stage.

The three important factors that determine thedevelopment of spacing of shear bands are: (a) therate of initiation, (b) the rate of growth (or the velo-city of shear bands), and (c) characteristic time ofinteraction between shear bands.

6.1. Initiation stage

Shear bands nucleate at the internal boundary ofthe specimen. The initiation sites are determinedby the small perturbations of initial deformation.They may be located at favorably oriented grainsand defects. Initiation of shear bands requires acritical strain at a certain stress. Heterogeneousmicrostructural or surface effects (boundarygeometry, defects, orientation of grains, etc.) deter-mine the range of strains in which the nucleationtakes places. We will treat the initiation like anucleation process. In heterogeneous nucleationtheory, the term embryo is used as a potentialnucleation site. The rate of nucleation of bands isassociated with a plastic strain rate, e.

The probability of nucleation is given byP(V0,S0), in a reference volume, V0, or surface, S0,depending whether initiation occurs in the bulk oron the surface. It can be described by a modifiedWeibull [46] distribution, using strain as the inde-pendent variable (in lieu of stress, in the conven-tional approach). Thus,

P(V0,S0) � 1� exp�� e�eie0�ei�q�, (36)

where ei is the critical strain below which noinitiation takes place; e0 is the average nucleationstrain (material constant); e is the variable; and q is

a Weibull modulus. The different Weibull modulireflect the dispersion of shear band nucleation. Fordifferent materials the nucleation curve can havedifferent shapes and positions, adjusted by settingq, ei, and e0. The rate of nucleation can be obtainedby taking the time derivative of the aboveexpression. In the geometry used in the experi-ments, the initiation occurs on the inside surfaceof the cylinder. For Ti and Ti–6Al–4V, the averagenucleation strains are selected as 0.4 and 0.12,respectively, to best fit the experimental results; qwas given values of 2, 3, 6, and 9, providing differ-ent distributions. Figs. 14(a) and (b) show the pre-dicted distributions of initiation strains for Ti andTi–6Al–4V, respectively.

There is also a continuing shielding effect, sothat the bands that actually grow can be a fractionof the total possible initiation sites. Fig. 15 showsthe schematic interaction between embryos andgrowing shear bands. Each growing band generatesa shielded region around itself, that is the resultof unloading.

The factors governing the evolution of self-organization include: (a) the strain rate, e; (b) thevelocity of growth of shear band, V; (c) the initialspacing, L.

Different scenarios emerge, depending on thegrowth velocity V. The shielded volume is depen-dent on the velocity of propagation of stressunloading and is directly related to the momentumtransfer used by Grady and Kipp [21]; this is givenby k1V, where k1�1. If the unloading velocity wereequal to the propagation velocity, one would have

Fig. 14. Probability of nucleation of shear bands in Ti (a) andTi–6Al–4V (b) as a function of shear strain for four values ofq: 2, 3, 6, and 9.


Fig. 15. Two-dimensional representation of concurrent nucleation and shielding.

k1 = 1. The activation of embryos at three times,t1, t2, and t3, is shown in Fig. 15. When V is low,the embryos are all activated before shielding canoccur, and the natural spacing L establishes itself.As V increases, shielding becomes more and moreimportant, and the number of deactivated embryosincreases. For a full nucleation case, the predictedspacing of shear bands in Ti is roughly 0.24 timesof the maximum length of shear bands. The widthof the unloaded region is 2k1 times the length ofthe shear band.

The shielding effect can be expressed by S:

S � 1�eL

k0k1V. (37)

The parameter k0 defines the range of strains overwhich nucleation occurs. It can be set as 2(e0�ei):

k0 � 2(e0�ei). (38)

The physical meaning of the shielding factor Sis determined by the ratio of two characteristictimes. One is the characteristic time for completenucleation, t:

t �k0

e�

2(e0�ei)e

. (39)

The other characteristic time is ts � t�tcr, wheretcr is a critical time at which complete shieldingoccurs:

tcr �L

k1V. (40)

It is assumed that the nuclei that have not beenactivated after tcr can no longer be initiated. There-fore, the shielding factor is defined as:

S �t�tcr

t. (41)

This expression correctly predicts an increase inshielding S with increasing V, decreasing e, anddecreasing L. For an extremely large velocity ofpropagation of shear band, the critical time is verysmall and the shielding factor is close to one,which means almost complete shielding. The prob-ability of nucleation under shear band shielding,P(L), is obtained by combining Eq. (37) and Eq.(36):

P(L) � (42)

�(1�S)·P(V0,S0) � � eL2(e0�ei)V

� 1� exp�� e�eie0�ei�q�� if tcr � t

P(V0,S0) � 1� exp�� e�eie0�ei�q�� if tcrt

When S = 0 no shielding effect exists and allnuclei grow. If, in the other extreme case, S = 1, nonucleation can happen. Fig. 16(a) shows predictedevolutions of nucleation probabilities as a functionof increasing strain, for different values of theshielding factor, S: 0, 0.2, 0.4, 0.6. This simplemodel has the correct physics and shows how theinitial distribution of activated embryos can beaffected by different parameters. Xue et al. [47]and Nesterenko et al. [48] present a preliminary,simplified version of this analysis. It explains,


Fig. 16. (a) Effect of shielding on the probability of nucleationas a function of plastic strain (different values of shielding, S,given in plot). Representative values for Ti–6Al–4V. (b) Prob-ability of nucleation of shear bands (incorporating calculatedshielding factor S) for Ti (q=6, S=0.14) and Ti–6Al–4V (q=4,S=0.89).

qualitatively, the smaller spacing of shear bands inTi, as compared to Ti–6Al–4V. Fig. 16(b) showsthe predictions for these two materials. ApplyingEq. (37), we obtain the shielding factors for thetwo cases. The following values were used: V=153and 556 m/s for Ti and Ti–6Al–4V, respectively;err = 1.75×104 s�1; ei�e0 = 0.1 and 0.07 for Ti andTi–6Al–4V, respectively; the range is numericallylarger in Ti than in Ti–6Al–4V. In comparison withexperimental observations, spacings L = 0.29 mmfor Ti and L = 0.1 for Ti–6Al–4V are obtainedfrom the Wright–Ockendon perturbation analysis

(Table 3). The ratio of the unloading velocity andthe velocity of shear band, k1, is taken as 0.2. It isrecognized that these values are only approximate.The resultant shielding factors S are 0.14 for Ti and0.89 for Ti–6Al–4V. Nevertheless, they provide agood estimate of the evolution of nucleation forthe two cases. Fig. 16(b) shows that the spacing ofshear bands is far from reaching saturation in Ti–6Al–4V due to significant shielding, whereas thiseffect is not so pronounced for Ti.

The shear-band spacing, corrected for shielding,is represented by:

Ls �LWO

P(L)�tt

�LWO

(1�S). (43)

LWO is the Wright–Ockendon spacing. This spacingis plotted as a function of S in Fig. 17(a) and as afunction of plastic strain, at a fixed value of S, inFig. 17(b). It is clear that the shear-band spacingdecreases with strain until a final, steady statevalue is reached (Fig. 17(b)). Using the calculatedshielding factors S of 0.14 for Ti and 0.89 for Ti–6Al–4V, the corresponding values for Ls are 0.34and 0.9 mm, respectively. These values onlyapproximate the experimental results (0.18 and0.53 mm, respectively) but they have the correcttrend, i.e. the spacing for Ti–6Al–4V is larger thanfor Ti. This eliminates the incorrect predictions ofWO and M models where the spacing betweenshear bands in Ti–6Al–4V is smaller than in Ti.Considering the uncertainties of experiments andmeasurements, it is felt that the agreement is satis-factory.

6.2. Growth stage

It was observed in experiments that growingshear bands form an approximately periodical pat-tern. This is called a self-organization phenom-enon.

Further deformation produces heterogeneousgrowth of these shear bands: some shear bandsgrow faster than others. The favorably loaded shearbands grow faster, while the unloaded shear bandsslow down, and finally stop. The ‘ living’ shearbands compete each other and construct a new spa-tial pattern by following the self-organization rule.The dead small shear bands may be left at their


Fig. 17. Predicted shear-band spacing, Ls, as a function of (a)shielding factor S at large strain; and (b) strain with constant S.

original locations, or may be merged into the largeplastic deformation of the surrounding area. In thelatter case the apparent number of shear bands iseffectively reduced.

The dominant factors in this process depend onthe material parameters, constitutive character,loading condition, and strain rate. The velocity ofshear bands, relative to unloading, is an importantfactor that affects their spatial distribution, anaspect which the current theories have not covered.In the extreme case when time of propagation ofshear band through the specimen is smaller than

the characteristic time of shear band interactions,no self-organization during growth will take place.

The driving force for shear-band propagation isthe release of elastic energy. The rate of nucleationis quite different from the growth rate (or growthvelocity). It is reasonable to assume that the growthis governed by stress, whereas initiation is gov-erned by strain. The necessary condition is:

tg � ti, (44)

where tg and ti are the critical shear stresses forgrowth and initiation, respectively. The greater thedifference, the higher the velocity of propagation.The unloading waves sweep through the surround-ing area of a shear band and make any newnucleation impossible within this area.

The interaction of shear bands leads to the com-petitive growth of the propagating shear bands. Thevolume of material shielded from furthernucleation and growth increases with the length ofa shear band in cylindrical geometry and is notconstant, as predicted with these elements in place,one can construct a more realistic two-dimensionaltheory for the evolution of the self-organization ofshear bands.

Different evolution stages for shear bands areschematically indicated in Fig. 18. At a certainlength li, the spacing is Li. The growth becomesunstable at a critical length lcr,i, and alternate shearbands grow with a new spacing Li+1; the othershear bands stop growing. The mathematical rep-resentation of the step function shown in Fig. 13 is

L � L0 � n

j � 1

k�j·H(l�lcr,j), (45)

where H(l�lcr,i) is a Heaviside function. The para-meters k�

j can be expressed as:

k�j � f(V,L0,ei,ep), (46)

where V is the shear band propagation velocity,L0 is the initial spacing, ei is the critical strain forinitiation and ep is the critical strain for propa-gation. A similar phenomenon was observed andanalyzed by Nemat Nasser et al. [49] in the fractureof glass. Thermal stresses were used to drive paral-lel cracks. As their length increased, their spacingwould also increase. This occurred in a discontinu-ous manner.


Fig. 18. (a) Schematic diagram of the evolution of shear-bandspacing at different levels: t1, random initiation; t2, self-organi-zation into ‘periodic’ pattern among nuclei; t3, some shearbands grow faster, suppressing others; t4, self-organization ofdeveloped shear bands. (b) Spacing of propagating shear bandsas a function of length.

7. Conclusions

Collective organization processes take placeduring the formation of shear bands. All the currenttheories on prediction of spacing of shear bands arebased on one-dimensional analyses of shear bands.Therefore, it is no surprise that the current theoriescannot explain the self-organization behavior ofshear bands when they grow into two and threedimensional patterns.

The evolution of multiple adiabatic shear bandswas investigated in commercially pure titaniumand Ti–6Al–4V alloy through a radial collapse

technique of a thick-walled cylinder under high-strain-rate deformation. Shear-band initiation,propagation, as well as spatial distribution wereexamined under different global strains. The shearbands nucleate at the internal boundary of thespecimens and construct a periodical distributionat an early stage. The shear bands undergo bifur-cation as they progress in their spiral trajectory andas their spacing increases. The shear bands are fav-ored initiation sites for failure, which occurs byvoid nucleation, growth, and coalescence inside thethermally softened regions. The evolution of themorphology of the voids is determined by therestrictions imposed by the bands.

The evolution of shear-band pattern during thedeformation process reveals a self-organizationcharacter. The differences of mechanical responsebetween the two alloys are responsible for signifi-cant differences in the evolution of the shear bandpatterns. The number of shear bands initiated in Ti(spacing of 0.18 mm) is considerably larger thanin Ti–6Al–4V (spacing of 0.53 mm); on the otherhand, the propagation velocity of the bands in Ti–6Al–4V (V�556 m/s) is approximately three timeshigher than in Ti (V�153 m/s). The propagationvelocities are successfully compared with valuespredicted from a variational analysis developed byMercier and Molinari [44]. The experimentallyobtained shear-band spacings are compared withtheoretical predictions by Grady and Kipp [21],Wright and Ockendon [22], and Molinari [23] andthe shortcomings of the predictions are discussed.The experimental results presented here for Ti andTi–6Al–4V corroborate the earlier results [20] thatself-organization is an important phenomenon indeformation by shear-band propagation. However,the shear-band interactions are more complex thanpreviously thought and their spacing cannot be pre-dicted by the one-dimensional perturbation theoriesof Wright and Ockendon [22] and Molinari [23].The Grady–Kipp theory cannot accommodate theincreased spacing as the shear-band size increases,since it is also one-dimensional. The proposed twodimensional model correctly describes the differ-ences of spacings between behaviors of Ti and Ti–6Al–4V. It incorporates elements that are out-lined below:


1. Rate of nucleation of shear bands. The prob-ability of nucleation, P(V0,S0), in a referencevolume V0, or surface S0 was successfullydescribed by a Weibull [46] distribution inwhich the stress was replaced by strain as theindependent variable. Parameters defining thedistribution are a critical strain for nucleation, amean nucleation strain, and a Weibull modulus.There can also be shielding at the nucleationstage, depending on the relative values of therate of nucleation and rate of growth.

2. Rate of growth, or velocity of propagation. Thisis an important factor in their self organization.Shear bands compete among themselves andgradually change their patterns. A ‘Darwinian’natural selection takes place, and a large numberof small bands evolves gradually into a smallernumber of large bands, due to the shielding ofstresses produced during growth. Such evol-ution of shear band pattern occurs under ahomogeneously distributed pressure acting onthe external boundary of the cylindrical speci-men. This is a typical self-organization processamong “aging” population of shear bands.

Acknowledgements

This work was supported by USA ArmyResearch Office under MURI Program No.DAAH004-96-1-0376 (Program Manager DrDavid Stepp). Discussions with and support by DrT.W. Wright (US Army Research Office) are gre-atly appreciated. Prof. V. Lubarda’s and Dr. D.Curran’s help are gratefully acknowledged.

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